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HW #14 - Homework-014 64 Assume that the lifetime of a...

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Homework-014 64. Assume that the lifetime of a certain component has an exponential distribution with mean 3. Find all of the following: the value of the parameter, f , the 55th percentile and the standard deviation of lifetimes, the probability that a component lasts longer than 7.5 years, the probability, for a component that has functioned for 1 year, that it will function for at least three more years. SOLUTION: Let X indicate the lifetime of a certain component, i.e. X Exp( λ = 1 μ = 1 3 ). a. Parameter, i.e. λ = 1 3 . b. f , the pdf, f ( x ) = 1 3 e x 3 , x 0. c. x . 55 = ln(1 p ) λ = ln(1 0 . 55) 1 3 = 2 . 396. d. P(X > 7 . 5) = e 1 3 (7 . 5) = e 2 . 5 (= 0 . 082). e. P(X 1 + 3 | X > 1) = P(X 3) = e 1 3 (3) = e 1 (= 0 . 368). 65. Do Exercise 6 on p.270. SOLUTION: Let X indicate the waiting time of a certain component, i.e. X Exp( λ = 1). a. P(X > 5) = e (1)(5) = e 5 (= 0 . 007). b. Based on a., 5 minutes is an unusual long time to wait (0.007 is too small). c. If one really waited for 5 minutes until the next hit, such unusual event happens most likely because the claim is wrong (i.e mean is supposed to be higher).
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